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Configurations and invariant nets for amenable hypergroups and related algebras


Author: Benjamin Willson
Journal: Trans. Amer. Math. Soc. 366 (2014), 5087-5112
MSC (2010): Primary 43A62; Secondary 43A07, 20N20
DOI: https://doi.org/10.1090/S0002-9947-2014-05731-0
Published electronically: July 16, 2014
MathSciNet review: 3240918
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Abstract: Let $ H$ be a hypergroup with left Haar measure. The amenability of $ H$ can be characterized by the existence of nets of positive, norm one functions in $ L^1(H)$ which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of $ H$ using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in $ L^1(H)$ which tend to left invariance weakly, but not in norm.

We define the semidirect product of $ H$ with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product.

These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra.


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Additional Information

Benjamin Willson
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Address at time of publication: Department of Mathematics, Hanyang University, Seoul, South Korea
Email: bwillson@ualberta.ca

DOI: https://doi.org/10.1090/S0002-9947-2014-05731-0
Keywords: Semidirect product, hypergroup, amenable, configurations, Lau algebra, invariant net, Reiter net
Received by editor(s): June 1, 2011
Received by editor(s) in revised form: October 22, 2011
Published electronically: July 16, 2014
Additional Notes: The financial assistance of NSERC in the form of a Postgraduate Scholarship(Doctoral) and funding through a Discovery Grant awarded to Professor Anthony To-Ming Lau is gratefully acknowledged.
Article copyright: © Copyright 2014 American Mathematical Society